We investigate the group irregularity strength (sg(G)) of graphs, i.e. the
smallest value of s such that taking any Abelian group \gr of order s,
there exists a function f:E(G)\rightarrow \gr such that the sums of edge
labels at every vertex are distinct. We prove that for any connected graph G
of order at least 3, sg(G)=n if n=4k+2 and sg(G)≤n+1 otherwise,
except the case of some infinite family of stars